Derivative of eᶜᵒˢˣ⋅cos(eˣ)

Let's now use what we know about the chain rule and the product rule to take the derivative of an even weirder expression. So, we're gonna take the derivative, we're gonna take the derivative of either the cosine of x times the cosine of e to the x. So, let's take the derivative of this. So, we can view this as the product of two functions. So, the product rule tells us that this is going to be the derivative with respect to x of e to the cosine of x, e to the cosine of x times cosine times cosine of e to the x plus, plus the first function, just e to the cosine of x. E to the cosine of x, times the derivative of the second function. Times the derivative with respect to x of cosine of e to the x. Cosine of e to the x. And so, we just need to figure out what these two derivatives are. And so you can imagine the chain rule might be applicable here. So, let me make it clear. This, we got from the product rule. Product, product rule. But then, to evaluate each of these derivatives, we need to use the chain rule. So, let's think about this a little bit. So, the derivative, let me copy and paste this so I don't have to rewrite it. So, copy and paste. So, let's think about what the derivative of e to the cosine of x is, e to the cosine of x. So, we can view our outer function as e to the something ,as e to the something and the derivative of e to the something with respect to something is just going to be e to that something so it's going to be e to the cosine of x. So, let me do that in that same blue color. So, it's going to be e, actually I'm gonna do it in that, actually I'm gonna do it in a new color. Let me do it in magenta. So, the derivative of e to the something with respect to something is just e to the something. It's just e to the cosine of x, and we have to multiply that times the derivative of the something with respect to x. So, what's the derivative of cosine of x with respect to x? Well, that's just negative sine of x. So, it's times negative sine of x. And so, we figured out this first derivative. Let me make it clear. This right over here is the derivative of e to the cosine of x. Derivative of e to the cosine of x, with respect to, with respect to cosine of x. And this right over here, this right over here is the derivative of cosine of x with respect, with respect to x. And we just took the product of the two. That's what the chain rule tells us. Fair enough. Now, let's figure out this derivative out here. So, we want to find the derivative with respect to x of cosine e to the x. So, once again, let me copy and paste it. So, we need to figure out this thing right over here. So first, just like we did, we're just going to apply the chain rule again. We need to figure out the derivative of cosine of something, in this case, e to the x, with respect to that something. So, this is going to be equal to derivative of cosine of something with respect to that something is equal to the negative sine of that something. Negative sine of e to x, of e to the x. Once again, we can view this as the derivative of cosine of e to the x, with respect to, with respect to e to the x. And then we multiply that times the derivative of the something with respect to x. So, let me do this, and this, and I'm running out of colors. Let me do this in the screen color. So, times the derivative of e to the x with respect to x is just e to the x. So that right over there is the derivative of e to the x with respect, with respect to x. And so, we're essentially done, we just have to substitute what we found using the chain rule back into our original expression. The derivative of this business up here is going to be equal to, let me just copy and paste everything just to make everything nice, nice and clean. So, copy and paste. So, that is going to be equal to, is going to be equal to this times cosine e to the x, so this is going to be, we'll see, we could put the e to the x out front, we could put the negative out front, so we could write it as negative e to the cosine x. e to the Cosine x, times Sine of x, times Sine of x, times Cosine of e to the x, times Cosine of e to the x. So, that's this first term here. Plus, plus e of the cosine x times all of this stuff. And so, let's see, we can put the negative out front again. So let's put that negative out front. So we have a negative, negative of e to the cosine of x times e to the x. I can write it this way. e to the x times e to the cosine of x, and you could simplify that, or combine it, since you're multiplying two things with the same base, but I'll just leave it like this, like this. e to the x, times e to the cosine of x, times the sine. We already have the negative, so then we have sine of e to the x. Sine of e to the x, right over here. Times sin, sin, of e to the x, we had negative sin e to the x times e to the x, negative sin of e to the x times e to the x, and then that multiplied by e to the cosine of x so we have the exact same thing right over here. And we're done.